Records and 2-block records of 1-dependent stationary sequences under local dependence

Abstract

We first study the records of several examples of 1-dependent stationary sequences $\{X_n\}$, some of them which satisfy a local independence condition introduced in Haiman (1987a), whereas the other do not satisfy this condition. For the second category (called with local dependence), we then extend a result of the above cited paper by proving, (under some regularity conditions on the joint distribution of $(X_1, X_2, X_3))$, that the 2-block record times of $\{X_n\}$ a.s. coincide, (via a translation of the time index and from some index on) with the record times of an appropriately constructed i.i.d. sequence $\{X_n\}$, whereas the 2-block record values of $\{X_n\}$ and the record values of $\{X_n\}$ are imbricate.